American Institute of Mathematical Sciences

Advanced
September  2021, 29(4): 2553-2560. doi: 10.3934/era.2021001

Note on coisotropic Floer homology and leafwise fixed points

Fabian Ziltener

Utrecht University, Mathematics Institute, Budapestlaan 6, 3584 CD Utrecht, The Netherlands

Received  November 2019 Revised  September 2020 Published  September 2021 Early access  January 2021

For an adiscal or monotone regular coisotropic submanifold $ N $ of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of $ N $. Given a Hamiltonian isotopy $ \varphi = ( \varphi^t) $ and a suitable almost complex structure, the corresponding Floer chain complex is generated by the $ (N, \varphi) $-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold.

Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This reproduces earlier results of mine.

The first construction also gives rise to a Floer homology for a Boothby-Wang fibration, by applying it to the circle bundle inside the associated complex line bundle. This can be used to show that translated points exist.

Keywords: Floer homology, coisotropic submanifold, leafwise fixed point, leafwise intersection, translated point.
Mathematics Subject Classification: Primary: 53D40.
Citation: Fabian Ziltener. Note on coisotropic Floer homology and leafwise fixed points. Electronic Research Archive, 2021, 29 (4) : 2553-2560. doi: 10.3934/era.2021001
References:
[1]

P. Albers, A note on local floer homology, arXiv: math/0606600. Google Scholar

[2]

P. Albers, A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology, Int. Math. Res. Not. IMRN, (2008), Art. ID rnm134, 56 pp. doi: 10.1093/imrn/rnm134.  Google Scholar

[3]

Yu. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J., 95 (1998), 213-226.  doi: 10.1215/S0012-7094-98-09506-0.  Google Scholar

[4]

K. Cieliebak, A. Floer, H. Hofer and K. Wysocki, Applications of symplectic homology, II, Stability of the action spectrum, Math. Z., 223 (1996), 27-45.  doi: 10.1007/BF02621587.  Google Scholar

[5]

A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom., 28 (1988), 513-547.  doi: 10.4310/jdg/1214442477.  Google Scholar

[6]

A. Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math., 41 (1988), 775-813.  doi: 10.1002/cpa.3160410603.  Google Scholar

[7]

A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys., 120 (1989), 575-611.  doi: 10.1007/BF01260388.  Google Scholar

[8]

H. Geiges and A. I. Stipsicz, Contact structures on product five-manifolds and fibre sums along circles, Math. Ann., 348 (2010), 195-210.  doi: 10.1007/s00208-009-0472-z.  Google Scholar

[9]

V. L. Ginzburg and B. Z. Gürel, Local Floer homology and the action gap, J. Symplectic Geom., 8 (2010), 323-357.  doi: 10.4310/JSG.2010.v8.n3.a4.  Google Scholar

[10]

V. L. Ginzburg and B. Z. Gürel, Fragility and persistence of leafwise intersections, Math. Z., 280 (2015), 989-1004.  doi: 10.1007/s00209-015-1459-y.  Google Scholar

[11]

A. Kapustin and D. Orlov, Remarks on $A$-branes, mirror symmetry, and the Fukaya category, J. Geom. Phys., 48 (2003), 84-99.  doi: 10.1016/S0393-0440(03)00026-3.  Google Scholar

[12]

C.-M. Marle, Sous-variétés de rang constant d'une variété symplectique, Astérisque, 107–108, Soc. Math. France, Paris (1983), 69–86.  Google Scholar

[13]

Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I., Comm. Pure Appl. Math., 46 (1993), 949-993.  doi: 10.1002/cpa.3160460702.  Google Scholar

[14]

Y.-G. Oh, Addendum to: "Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I.", Comm. Pure Appl. Math., 48 (1995), 1299-1302.   Google Scholar

[15]

Y.-G. Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices, (1996), 305–346. doi: 10.1155/S1073792896000219.  Google Scholar

[16]

Y.-G. Oh, Localization of Floer homology of engulfed topological Hamiltonian loop, Commun. Inf. Syst., 13 (2013), no. 4, 399–443. Google Scholar

[17] Y.-G. Oh, Symplectic Topology and Floer Homology, Vol. 2, Floer homology and its applications, New Mathematical Monographs, 29, Cambridge University Press, Cambridge, 2015.   Google Scholar
[18]

M. Poźniak, Floer homology, Novikov rings and clean intersections, Northern California Symplectic Geometry Seminar, 119–181, Amer. Math. Soc. Transl. Ser. 2, 196, Adv. Math. Sci., 45, Amer. Math. Soc., Providence, RI, 1999. doi: 10.1090/trans2/196/08.  Google Scholar

[19]

S. Sandon, A Morse estimate for translated points of contactomorphisms of spheres and projective spaces, Geom. Dedicata, 165 (2013), 95-110.  doi: 10.1007/s10711-012-9741-1.  Google Scholar

[20]

F. Ziltener, Coisotropic submanifolds, leaf-wise fixed points, and presymplectic embeddings, J. Symplectic Geom., 8 (2010), 95-118.  doi: 10.4310/JSG.2010.v8.n1.a6.  Google Scholar

[21]

F. Ziltener, A Maslov map for coisotropic submanifolds, leaf-wise fixed points and presymplectic non-embeddings, arXiv: 0911.1460. Google Scholar

[22]

F. Ziltener, Leafwise fixed points for $C^0$-small Hamiltonian flows, Int. Math. Res. Not. IMRN, (2019), 2411–2452. doi: 10.1093/imrn/rnx182.  Google Scholar

show all references

References:
[1]

P. Albers, A note on local floer homology, arXiv: math/0606600. Google Scholar

[2]

P. Albers, A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology, Int. Math. Res. Not. IMRN, (2008), Art. ID rnm134, 56 pp. doi: 10.1093/imrn/rnm134.  Google Scholar

[3]

Yu. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J., 95 (1998), 213-226.  doi: 10.1215/S0012-7094-98-09506-0.  Google Scholar

[4]

K. Cieliebak, A. Floer, H. Hofer and K. Wysocki, Applications of symplectic homology, II, Stability of the action spectrum, Math. Z., 223 (1996), 27-45.  doi: 10.1007/BF02621587.  Google Scholar

[5]

A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom., 28 (1988), 513-547.  doi: 10.4310/jdg/1214442477.  Google Scholar

[6]

A. Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math., 41 (1988), 775-813.  doi: 10.1002/cpa.3160410603.  Google Scholar

[7]

A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys., 120 (1989), 575-611.  doi: 10.1007/BF01260388.  Google Scholar

[8]

H. Geiges and A. I. Stipsicz, Contact structures on product five-manifolds and fibre sums along circles, Math. Ann., 348 (2010), 195-210.  doi: 10.1007/s00208-009-0472-z.  Google Scholar

[9]

V. L. Ginzburg and B. Z. Gürel, Local Floer homology and the action gap, J. Symplectic Geom., 8 (2010), 323-357.  doi: 10.4310/JSG.2010.v8.n3.a4.  Google Scholar

[10]

V. L. Ginzburg and B. Z. Gürel, Fragility and persistence of leafwise intersections, Math. Z., 280 (2015), 989-1004.  doi: 10.1007/s00209-015-1459-y.  Google Scholar

[11]

A. Kapustin and D. Orlov, Remarks on $A$-branes, mirror symmetry, and the Fukaya category, J. Geom. Phys., 48 (2003), 84-99.  doi: 10.1016/S0393-0440(03)00026-3.  Google Scholar

[12]

C.-M. Marle, Sous-variétés de rang constant d'une variété symplectique, Astérisque, 107–108, Soc. Math. France, Paris (1983), 69–86.  Google Scholar

[13]

Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I., Comm. Pure Appl. Math., 46 (1993), 949-993.  doi: 10.1002/cpa.3160460702.  Google Scholar

[14]

Y.-G. Oh, Addendum to: "Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I.", Comm. Pure Appl. Math., 48 (1995), 1299-1302.   Google Scholar

[15]

Y.-G. Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices, (1996), 305–346. doi: 10.1155/S1073792896000219.  Google Scholar

[16]

Y.-G. Oh, Localization of Floer homology of engulfed topological Hamiltonian loop, Commun. Inf. Syst., 13 (2013), no. 4, 399–443. Google Scholar

[17] Y.-G. Oh, Symplectic Topology and Floer Homology, Vol. 2, Floer homology and its applications, New Mathematical Monographs, 29, Cambridge University Press, Cambridge, 2015.   Google Scholar
[18]

M. Poźniak, Floer homology, Novikov rings and clean intersections, Northern California Symplectic Geometry Seminar, 119–181, Amer. Math. Soc. Transl. Ser. 2, 196, Adv. Math. Sci., 45, Amer. Math. Soc., Providence, RI, 1999. doi: 10.1090/trans2/196/08.  Google Scholar

[19]

S. Sandon, A Morse estimate for translated points of contactomorphisms of spheres and projective spaces, Geom. Dedicata, 165 (2013), 95-110.  doi: 10.1007/s10711-012-9741-1.  Google Scholar

[20]

F. Ziltener, Coisotropic submanifolds, leaf-wise fixed points, and presymplectic embeddings, J. Symplectic Geom., 8 (2010), 95-118.  doi: 10.4310/JSG.2010.v8.n1.a6.  Google Scholar

[21]

F. Ziltener, A Maslov map for coisotropic submanifolds, leaf-wise fixed points and presymplectic non-embeddings, arXiv: 0911.1460. Google Scholar

[22]

F. Ziltener, Leafwise fixed points for $C^0$-small Hamiltonian flows, Int. Math. Res. Not. IMRN, (2019), 2411–2452. doi: 10.1093/imrn/rnx182.  Google Scholar

[1]

Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297
[2]

Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017
[3]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709
[4]

Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271
[5]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692
[6]

Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979
[7]

Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381
[8]

Sonja Hohloch. Transport, flux and growth of homoclinic Floer homology. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3587-3620. doi: 10.3934/dcds.2012.32.3587
[9]

Alexander Fauck, Will J. Merry, Jagna Wiśniewska. Computing the Rabinowitz Floer homology of tentacular hyperboloids. Journal of Modern Dynamics, 2021, 17: 353-399. doi: 10.3934/jmd.2021013
[10]

Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467
[11]

Teck-Cheong Lim. On the largest common fixed point of a commuting family of isotone maps. Conference Publications, 2005, 2005 (Special) : 621-623. doi: 10.3934/proc.2005.2005.621
[12]

Mircea Sofonea, Cezar Avramescu, Andaluzia Matei. A fixed point result with applications in the study of viscoplastic frictionless contact problems. Communications on Pure & Applied Analysis, 2008, 7 (3) : 645-658. doi: 10.3934/cpaa.2008.7.645
[13]

Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248
[14]

Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273
[15]

Ruhua Wang, Senjian An, Wanquan Liu, Ling Li. Fixed-point algorithms for inverse of residual rectifier neural networks. Mathematical Foundations of Computing, 2021, 4 (1) : 31-44. doi: 10.3934/mfc.2020024
[16]

Mark S. Gockenbach, Akhtar A. Khan. Identification of Lamé parameters in linear elasticity: a fixed point approach. Journal of Industrial & Management Optimization, 2005, 1 (4) : 487-497. doi: 10.3934/jimo.2005.1.487
[17]

Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, Sandro Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793
[18]

Grzegorz Graff, Piotr Nowak-Przygodzki. Fixed point indices of iterations of $C^1$ maps in $R^3$. Discrete & Continuous Dynamical Systems, 2006, 16 (4) : 843-856. doi: 10.3934/dcds.2006.16.843
[19]

Hans Koch. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 881-909. doi: 10.3934/dcds.2004.11.881
[20]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4927-4962. doi: 10.3934/dcdsb.2020320

2020 Impact Factor: 1.833

Tools

Metrics

  • PDF downloads (254)
  • HTML views (420)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy